Question 5 of 5: Short-Answer Machines and Power-Systems Theory

Question 5: Short-Answer Machines and Power-Systems Theory (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Nine short-answer parts on machines and power-systems fundamentals, each addressed in turn below.

(a) Delta-connected motor, one supply leg opens. A running motor keeps turning, though degraded. With one supply line open, the delta stator winding remains a closed loop: two of the three windings are directly across the two still-energized supply lines, and the third winding is connected across the same two lines in series through the other two windings, so it still carries current and still contributes torque — the machine does not simply lose one-third of its torque, it becomes an unbalanced single-phase-fed (open-phase) machine. It can therefore continue running on inertia at reduced torque capability, but with markedly unbalanced, elevated winding currents (the two directly-fed windings especially) that overheat the motor if the condition persists, and it cannot develop starting torque from standstill on the remaining two lines alone. Loss-of-phase protection (thermal or current-unbalance relays) is standard practice for exactly this failure mode.

(b) Generator droop. Speed (or frequency) droop is the deliberate, small negative slope built into a generator's governor characteristic: output frequency falls slightly as the generator's real-power output rises, rather than holding frequency perfectly flat. It is defined and calculated as $$R=\frac{\Delta f/f_{rated}}{\Delta P/P_{rated}}\times100\%=\frac{f_{NL}-f_{FL}}{f_{rated}}\times100\%$$ (no-load frequency minus full-load frequency, as a percentage of rated frequency, typically 2–5% for utility governors). Droop lets two or more generators share a common bus and divide load automatically in proportion to their capacity without hunting or fighting each other for control of frequency — a flat (zero-droop) governor on more than one machine on the same bus is unstable, since each machine would try to hold frequency exactly at its own local set point.

(c) Why a synchronous motor is not self-starting. At standstill the rotor's dc (or permanent-magnet) field is stationary while the stator's rotating field sweeps past it at synchronous speed; the instantaneous torque this produces alternates in direction every half-cycle as the rotating field alternately approaches and passes the rotor poles, so the average torque over one electrical cycle is zero. The rotor's inertia prevents it from accelerating fast enough to lock into synchronism from rest. Starting therefore needs some other means to bring the rotor near synchronous speed first — typically squirrel-cage/amortisseur (damper) bars for induction-type starting, or an external pony motor / variable-frequency drive — after which the dc field is applied and the rotor pulls into synchronous lock, from which point onward net unidirectional torque exists.

(d) Five specifications for selecting an induction motor. (1) Rated power/horsepower together with rated voltage and frequency; (2) synchronous speed (pole count) and full-load slip/speed; (3) starting torque and starting (locked-rotor) current, i.e. NEMA design class (A/B/C/D); (4) service factor, insulation class and temperature rise/duty cycle; (5) enclosure and environment rating (TEFC, explosion-proof, IP rating) together with efficiency class (e.g. NEMA Premium / IE3).

(e) Why low power factor draws a penalty. Generation, transmission lines, transformers and switchgear are all sized by apparent power (kVA/current capacity), while the customer is billed for real power (kW). A low-p.f. customer draws more current — and therefore consumes more of the utility's fixed kVA capacity and causes more I²R line loss — for the same billed kWh than a high-p.f. customer. The power-factor penalty recovers that unrecompensed infrastructure and loss cost, and gives the customer a financial incentive to install power-factor correction (capacitor banks or, as in Question 4, an over-excited synchronous motor).

(f) Can the 18 kVA, 20 kV/480 V, 60 Hz transformer safely supply 15 kVA at 415 V, 50 Hz? Check both limits, not just current. $$\begin{aligned} \frac{V}{f}\bigg|_{rated}&=\frac{480}{60}=8.00\ \text{V/Hz}\\ \frac{V}{f}\bigg|_{new}&=\frac{415}{50}=8.30\ \text{V/Hz}\ (+3.75\%) \end{aligned}$$ $$\begin{aligned} I_{rated}&=\frac{18{,}000}{480}=37.5\ \text{A}\\ I_{new}&=\frac{15{,}000}{415}=36.1\ \text{A}\ (\text{below }I_{rated}) \end{aligned}$$ The current is comfortably inside the nameplate rating, but core flux density tracks volts-per-hertz (B ∝ V/f), and operating at 8.30 V/Hz versus the rated 8.00 V/Hz over-fluxes the core by 3.75%. Answer: not strictly — it is current-safe but slightly over-fluxed. A transformer should not be run above its rated V/Hz: doing so drives the core toward saturation, sharply increasing exciting current, core (hysteresis + eddy) losses and localized heating, independent of how light the load current is. A conservative "safe" answer is no without first confirming the manufacturer's over-excitation margin; the current rating alone does not clear it.

(g) Why induction-motor efficiency is poor at high slip. The power crossing the air gap, P_ag, splits between rotor copper loss and mechanical output in fixed proportion to slip: P_rotor,cu = sP_ag and P_mech = (1−s)P_ag. As slip rises — heavy overload, or especially at starting where s = 1 — a progressively larger share of the air-gap power is dissipated as rotor I²R heating rather than converted to shaft work, so the theoretical efficiency ceiling (1−s) collapses toward zero exactly as slip approaches unity.

(h) Three causes of power-system harmonics and their effects. (1) Non-linear loads — variable-frequency drives, rectifiers and switch-mode power supplies — draw non-sinusoidal current from a sinusoidal voltage. (2) Transformer and machine core saturation, whose non-linear B–H curve produces harmonic-rich magnetizing current even from a pure sinusoidal supply voltage. (3) Arc-based loads (arc furnaces, arc welders) and electronic ballasts/LED drivers. Effects: extra I²R heating in transformers, motors and especially neutral conductors (triplen harmonics add arithmetically in a 4-wire neutral instead of cancelling); resonance between line/cable capacitance or power-factor-correction capacitor banks and system inductance, causing harmonic overvoltage; nuisance tripping/mis-operation of protective relays and metering; and interference with communication and control circuits.

(i) Why the transformer core is laminated, and three causes of core damage. Laminating the core into thin, mutually insulated steel sheets breaks up the low-resistance paths that would otherwise carry large circulating eddy currents; eddy-current loss scales with the square of lamination thickness (P_e ∝ t²f²B_max²), so thin laminations cut this loss (and the resulting heating) dramatically compared with a solid core. Three causes of damage to the core: (1) inter-laminar insulation breakdown (from ageing, overheating, or mechanical damage during assembly), which restores a low-resistance path and produces a local eddy-current hot spot; (2) over-excitation — overvoltage or under-frequency operation — driving the core into saturation with excessive flux and heating (exactly the mechanism examined in part (f)); (3) mechanical looseness of the core clamping/stacking bolts, letting laminations vibrate and chafe against each other, wearing through the inter-laminar insulation over time (through-fault currents and their associated mechanical/thermal stress are a related aggravating cause).